neither, in my opinion. an invention certainly is something which did not exist before, and pi's value is a property of space, i'd say, so it existed since the big bang.a discovery is something existing you found without knowing it existed before. everybody knew the relation between a circle's diameter and circumference existed, it only remained to determine its concrete value.

That the circumference of a circle has a constant ratio to its diameter was a discovery, albeit a very ancient one. Approximate values for this constant were later discovered, with increasing accuracy. The symbol "π" was assigned to this constant, though it was not invented for that purpose, already being a Greek letter. The fact that π is irrational was also discovered much later, and that it was transcendental, later still.

Basically, every piece is another discovery, so I don't think it's unreasonable to say in a loose sense that we "discovered π". It certainly doesn't make sense to say that we invented it.

However, the mathematical notion of a circle is an abstraction. A set of infinitely small points that are the same distance from a center. The fact that these points are infinitely small creates the need for a transcendental number to express the ratio to the radius. That construct does not exist in nature, only in the system created by humans.

danlovy wrote:However, the mathematical notion of a circle is an abstraction. A set of infinitely small points that are the same distance from a center. The fact that these points are infinitely small creates the need for a transcendental number to express the ratio to the radius. That construct does not exist in nature, only in the system created by humans.

Not really. Take a solid cube. Divide one of its angles into sevenths. Now mark one of the points of intersection. You now have another transcendental length.

These don't really result from particular arrangements of points; these numbers must exist in a continuum. Now, we may or may not have invented the continuum, but once the idea was there, we discovered many surprising facts about it.

Let me put it another way. Suppose I right a short story. I then count the number if words, and discover that my story is 1001 words long. I certainly didn't invent this constant; I invented the story, then investigated it to see what I could discover. This seems like a reasonable analogy to mathematics.

As for calculus, that's a method, so of course it was invented. But again, many facts regarding the method are necessary consequences which were but generally known when it was first developed. So those were discoveries.

I'm of the opinion that axioms and definitions are invented, and theorems are discovered by exploring their implications. Once you define a circle in Euclidean space, and you have a concept of real numbers, it can be proven that the circumference:diameter ratio of a circle is a real constant (defined as pi), and that there are many other expressions equal to this same exact number.

That is why it is just a question of definitions. One might be sympathetic to the above position, but if you are using "A discovery is a truth that is independent of human existence" pi is not a discovery, because of the dependency going back to human modeling activity. All abstract objects are inventions under this definition.

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

If I create an abstract, internally consistent world, I can 'discover' things about that world. Remember the whole flat land universe? It's a 2D abstraction where all the physics occur in two dimensions. It does not map to a real world, but it has things that can be 'discovered'.

Circles are an abstractions that do approximate things in the real world so 'discoveries' in this abstract context have value.

This is an argument about angels dancing on pin. However, I've been running into much pseudo-science (and many pseudo-scientists) these days and these lines are very blurred.

I think it's arguable that there really is no such thing as invention at all. There is a conceptual space of possibilities, and all we can do is discover them.

I was thinking about this in terms of copyright the other day. You can't copyright a number, because numbers are discovered, not created. But pretty much any creative work can be encoded as a (really long) number. Which can't be copyrighted, right? So you can distribute that really long number freely, right?

More familiar than that is patents on algorithms. Lots of people argue that algorithms should not be patentable because they are mathematical methods and those are discoveries, not inventions. But one can view the world such that pretty much everything is algorithms, the whole world is a complex bit of math that we're embedded in any everything we "invent" is just discovering something that can be done with it. So if everything is algorithms then nothing is patentable, right?

In both of those cases it seems like it's really just the complexity that marks the boundary of our distinction between "discovery" and "invention".

Also possibly marking that boundary is... what I want to call "centrality", or maybe "importance". You can discover that there is some funny irrelevant largely unimportant pattern to numbers or shapes, or equivalently invent a game to play with numbers or shapes exploiting that pattern (e.g. find the next whatever that fits the pattern). But if you discover the Fundamental Theorem of Whatever, nobody's going to mistake that for invention of some recreational mathematics game. That's a real important discovery that's fundamental to all of whatever, not just some dumb thing someone made up for fun.

doogly wrote:That is why it is just a question of definitions. One might be sympathetic to the above position, but if you are using "A discovery is a truth that is independent of human existence" pi is not a discovery, because of the dependency going back to human modeling activity. All abstract objects are inventions under this definition.

But that is not in fact what "discovery" means, unless physical anthropologists have been using the word very wrong. A discovery is just something you learn that you didn't know before.

Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.---If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

In response to the notion that all inventions are just discoveries about the consequences of certain decisions, I would posit the following definition.

An invention is a simultaneous discovery of a set of decisions and of the fact that those decisions result in a desired consequence.

For the purposes of debate about Pi, "discovery" should then be taken to mean a discovery that isn't an invention.

So if you devise an algorithm for factoring large numbers, that's an invention. If you devise the game of Chess, you have invented Chess. If someone else then devises the Queen's Gambit as a useful strategy, they have invented the Queen's Gambit. If someone studies endgames and finds that a King+Bishop is insufficient to checkmate a lone enemy King, that's a discovery and not an invention.

If you devise a set of axioms for modelling space, that's an invention. When you later find that those axioms imply the existence of a constant relating circumferences and diameters, that's a discovery. If instead you preconceived the notion of such a constant and selected your axioms specifically to ensure its existence, then it could be considered an invention. But that's not what happened in the case of Pi.

So say someone discovers something, in your not-invention sense. Later, someone else notices that that discovery alone is useful to some desired end. Has the latter person invented something, simply by realizing an application for an earlier discovery? Or do the discovery and the later application together constitute an invention?

Pfhorrest wrote:So say someone discovers something, in your not-invention sense. Later, someone else notices that that discovery alone is useful to some desired end. Has the latter person invented something, simply by realizing an application for an earlier discovery? Or do the discovery and the later application together constitute an invention?

In my opinion, no. I think the concept of 'invent' vs. 'discover' only makes sense in the context of human intent, at least when you're digging down to philosophical edge cases. For most colloquial purposes, we have the concept of accidentally inventing something, and for practical purposes that should often still be patentable. But for the purposes of philosophical distinctions, I think to consider something a true invention, the discovery and the intent behind it have to go hand in hand.

Once you pick the appropriate set of definitions and axioms, I'd say that it's a "discovery" within that system to find out that the ratio of circumference to diameter is a constant value. If you use a different system, then there will be different discoveries within that system (for example, the locus that defines a circle under the Pythagorean metric gives you a square under the Manhattan metric, in which case "pi" equals 4). Pi happens to be a kind of interesting transcendental number just because it (or values closely related to it) appear in a bunch of apparently unrelated contexts - something which is itself a discovery of sorts. So you can "discover" that the area underneath exp(-x^2) is a rational multiple of the circumference-diameter ratio.

arbiteroftruth wrote:So if you devise an algorithm for factoring large numbers, that's an invention.

So...I've devised an algorithm for determining whether large integers are even or odd:-> (Look at the last digit in base ten. Is it in the set {0,2,4,6,8}? Even. Not? Odd.) --> Invention?

I've stumbled upon a property of large integers:-> The even ones yield a result whose last digit in base ten is in the set {0,2,4,6,8}. Odd ones don't. --> Discovery?

I've come up with a way of inventing things:-> Take a discovery and say something like "I've come up with an algorithm to determine whether {discovery}." --> Profit?

Jose

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Chimpanzees have been shown to take blades of grass, push them into ant hills and pull out delicious ants. Did it discover or invent this? Doesn't matter, it is a product of thought.

Is all discovery and invention the product of thought? The light bulb can be viewed as both invention (combining materials to accomplish a goal) and discovery (the properties of certain materials, when combined appropriately) has a new property - light.

Chimpanzees have been shown to take blades of grass, push them into ant hills and pull out delicious ants. Did it discover or invent this? Doesn't matter, it is a product of thought.

Is all discovery and invention the product of thought? The light bulb can be viewed as both invention (combining materials to accomplish a goal) and discovery (the properties of certain materials, when combined appropriately) has a new property - light.

Plenty of actions are products of thought that would not qualify as either inventions or discoveries. Practically all of them, in fact. When I decide to scratch my nose, I haven't invented or discovered anything. You are just taking a much larger set in order to discard any relevant distinctions between subsets.

OK but what if when you scratch, you sneak a corner of finger nail up into the nostril. You know, to investigate. Then maybe you discover something! Always be on the lookout to broaden the vistas of the known.

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

I would say that the number itself is not illegal, but that it could be used to do illegal things. If I learned that X can be used to make illegal copies of a DVD, then I have not broken copyright law. If I make illegal copies, then I have broken copyright law. If I tell other people that the number X can be used to make illegal copies, then I am teaching people how to break copyright law, which is itself an illegal action.

I am really surprised that no one has posted these videos yet*. If anyone cares, I am a realist. I get around the problem pointed out in the videos by saying that we do causally interact with non-physical objects.

A Mad Lib that Proves my Point wrote:I can pick a [thing] and claim it is mine. It is my [thing]. I can change it by [operation]. The result is still my [thing], although is now not identical to how it was before [operation]. Therefore, I have causally interacted with it. This is very important; my [thing] existed before I claimed ownership of it. It is like there was an machine that was turning out so many [things] that there is practically an endless supply of identical [things]. Because they are identical, [operation] will change every identical [thing] in exactly the same way. This allows me to make meaningful statements about the [things] that I do not own.

I could fill in the blanks with 'number' and 'adding 5', 'function' and 'integrating', or even 'pocket knife' and 'opening it'. This video is kind of like what I believe, but not exactly.

That numberphile video is talking about something different then I am.

All the data on that DVD? That's one big number. It's illegal to share that big number because that would be copying copyrighted data. Yet, on paper, you can't copyright a number because it is discovered, not created.